which once substituted back into Eq. () lead to an expression of \(\mathcal {H}_{\mathrm {grav}}^{(L)}\)  different from that of \(\mathcal {H}_{\mathrm {grav}}^{(E)}\)  in standard LQC (see [1]} for more details). Correspondingly,
one is able to obtain the following “effective" Hamiltonian [1]}, [3]},
\(\mathcal {H}^{\scriptscriptstyle {\mathrm {I}}}_{\text{eff.}}&=&\frac{3v}{8\pi G\lambda ^2}\Big \lbrace \sin ^2(\lambda b)-\frac{(\gamma ^2+1)\sin ^2(2\lambda b)}{4\gamma ^2}\Big \rbrace \nonumber \\&& ~~~~~~~~~~~~ +\mathcal {H}_M.\) 
